Figure out if you have a perennial or annual. Annuals you can control by mowing, you chope them off before they make seeds. If you let them seed the herbicide is unlikely to kill the seeds and a survival trait of annual grasses are the seeds often germinate at different times. Some of those seeds can last years before they grow.

Perennials are a lot harder. Perennial Rye and Bunch Grass can have roots a foot deep. Crabgrass can have eight-inch roots. The herbicide kills off the top, the deep roots regrow the grass. Even digging it out can be iffy. Repeated application of herbicide can eventually wear down the roots ability to re-propogate the grass maybe.
Rye isn't necessarily bad, it is decent forage. What makes Clover desirable is it is sweet.
One thing I do is plant Rape, it grows taller than the grasses and eventually chokes them out. Winter Rape is especially good at this. And winter rape really draws the Deer, better than Clover. When the winter Rape grows up to around 6-8 inches and then goes dormant for the winter, it gets really sweet when frost hits. Deer love that sweet taste. And Deer seem to favor full-grown Rape, even eat the seeds for the oil in the fall. And tall Rape makes a good spot to drop Fawns, much like tall grass or weeds.